14 problems

Let $A=(A,\oplus,\odot)$ be a finite skew brace. Its additive group is $A_{\oplus}$ and its multiplicative group is $A_{\odot}$. Smoktunowicz–Vendramin conjecture. If $A_{\oplus}$…

Let $m\geq 3$, and let $q(4m)$ denote the number of braces $B$ whose multiplicative group $(B,\circ)$ is a generalised quaternion group of order $4m$. Guarnieri–Vendramin's conject…

A skew brace is a set with two group structures, an additive group $(A,+)$ and a multiplicative group $(A,\circ)$, satisfying the skew brace compatibility law. A skew brace is of s…

Let $N$ be a finite soluble group, and write … A subgroup of $\operatorname{Hol}(N)$ is regular when its natural action on $N$ is free and transitive. Byott's conjecture. Every reg…

A skew brace $(A,+,\circ)$ is a set with additive and multiplicative group structures satisfying the skew left distributive law; its additive group is soluble if it has a finite de…

Nilpotent-type skew brace conjecture. The group $G$ is isomorphic to the multiplicative group of a skew left brace of nilpotent type.

Solvability conjecture. If $(G,N)$ is realizable and $G$ is insolvable, then $N$ is also insolvable.

Byott's conjecture. If $(G,N)$ is realizable and $G$ is insolvable, then $N$ is also insolvable.

Let $N$ be a finite soluble group, and let $G$ be a regular subgroup of the holomorph … of $N$, meaning that $G$ acts regularly on the underlying set of $N$. Regular-subgroup solub…

Let a quaternion brace be a brace whose multiplicative group is isomorphic to a generalized quaternion group. Power-of-two quaternion-brace conjecture. There are seven isomorphism…

For $m\in\mathbb{N}$, let $Q_{4m}$ be the generalized quaternion group, and call a brace a quaternion brace if its multiplicative group is isomorphic to a generalized quaternion gr…

Let $b(n)$ denote the number of non-isomorphic left braces of size $n$. Let $p,q$ be prime numbers such that $p<q$ and $q\not\equiv1\bmod p$. The order-$p^2q$ counting conjecture.…

Let $b(n)$ denote the number of non-isomorphic left braces of size $n$, and let $p>3$ be a prime. The order-$9p$ counting conjecture. … The conjecture is suggested by the table of…

Let $b(n)$ denote the number of non-isomorphic left braces of size $n$. Let $p>3$ be a prime. The order-$4p$ counting conjecture. … The statement is presented as a conjecture sugge…